Clearly, studies for evolution equations on the Heisenberg group are not trivial generalizations from those in the Euclidean space due to the fact that classical analyses (e.g. the Fourier transform for \(L^2(\mathbb {R}^n)\)  functions) do not work well. One may see two typical examples which are the free wave equation from [1]}, [2]}, and the linear damped wave equation [3]}, [4]} in the framework on \(\mathbf {H}_n\) . At this time, we would like to understand
